{numerical-linear-algebra} questions involving algorithms for linear algebra computations.

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2
votes
0answers
155 views

Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...
3
votes
1answer
78 views

Conjugate gradient algorithm where first search direction is not equal to residual

In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact ...
7
votes
1answer
177 views

Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
7
votes
1answer
166 views

Who first observed that Conjugate Gradient for Symmetric Positive Definite linear systems is a Krylov method?

Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived ...
10
votes
0answers
192 views

What are the difficulties in proving almost-everywhere stability of Gaussian elimination?

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
4
votes
1answer
145 views

standard practice for large dense truncated svd computations?

What are the standard methods of computing the rank-k truncated SVD of large dense matrices? My literature search yields results only for large sparse matrices. I assume for k small that you use a ...
2
votes
0answers
51 views

Most efficient algorithm for computing norm of the residual for the least squares problem in the rank deficient case

I have a large $m\times n$ data matrix $A$, $m>n$, and response $m$-vector $b$. I need to calculate $E = ||Ax-b||_2$ as quickly as possible, where $x$ is the least squares solution. I don't need ...
4
votes
1answer
361 views

Norm of inverse confluent Vandermonde matrix

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} ...
2
votes
0answers
146 views

Checking for error in conjugate gradient algorithm

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
0
votes
1answer
129 views

Nonlinear matrix equation 2

Solve the following nonlinear equations for $v$ and $w$ $Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$ $Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$ $v^Tw=w^Tv=0$ $v^Tv=w^Tw=1$ where $\lambda_1, \lambda_2, ...
3
votes
1answer
761 views

Nonlinear matrix equation

Solve the following nonlinear equations for $v$ and $w$ $Avv^TAw=\lambda_1v+\lambda_2w$ $Aww^TAv=\lambda_1w+\lambda_2v$ $v^Tw=w^Tv=0$ $v^Tv=w^Tw=1$ where $\lambda_1, \lambda_2, \lambda_3$ are ...
3
votes
0answers
423 views

Quantifying the failure of the Cholesky factorization test for indefinite matrices

The Cholesky factorization is the classic test to check if a matrix is positive definite. In infinite precision it is also an exact test: A matrix has a Cholesky factorization iff it is positive ...
7
votes
2answers
1k views

Finding the smallest eigenvalues of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $M$. $M$ is a Laplacian matrix, and it has the following structure: ...
0
votes
1answer
169 views

Ease of calculation of norm

I have SPD matrix A and two vectors z and b. Is there exist a norm where I can calculate $||A^{1/2}b-z||$ without having to calculate $A^{1/2}b$ explicitly ?
5
votes
0answers
133 views

reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then $$ \|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2). $$ ...
0
votes
0answers
125 views

Finding peaks and determining noise

Hello , Im having one matrix which is product of two FFT transforms of one fits image ( astronomical image ). In that matrix you could find 3 peaks. One largest in center, and two around central ...
2
votes
2answers
154 views

Inflate a simplex, change rows to make the rank n

I have a simplex, n + 1 points in $\mathbb{R}^n$, which may have rank $r < n$. Is there a cheap way of "inflating" it to rank $n$, changing a few, all but $r$, of the points ? The points are ...
2
votes
1answer
300 views

A question for solutions of perturbed linear systems

Consider a linear system $$Ax=b\qquad (*)$$ and a sequence of perturbed linear systems $$(A+\delta A_n)x=b+\delta b_n. \qquad (n)$$ Suppose that all the linear systems are consistent (i.e., ...
3
votes
0answers
120 views

Computing the norm of the columns of an implicitly defined matrix

I have an $n \times n$ matrix $M = \Sigma W$ where $\Sigma$ is diagonal and $W$ orthogonal. $W$ is implicitly defined, i.e. I can only perform matrix-vector products (but I also have access to $W^T$). ...
6
votes
3answers
3k views

Interesting relationships between Cholesky decomposition and diagonalization

Let $\Sigma$ be a hermitian positive definite matrix and $L$ be it's Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ ...
1
vote
2answers
2k views

Low rank Matrix factorization

Hello, I've a SPD matrix A; which needs to be factorized as ${A=SS^{T}}$. But, using Cholesky for this purpose is prohibitive in terms of computational cost. Moreover, matrix is Dense and has a slow ...
3
votes
0answers
186 views

Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with $m-$dimensional coordinates in ...
7
votes
2answers
473 views

polynomials with minimal $L_\infty$ norm on multiple disjoint intervals

It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
2
votes
0answers
490 views

How many iterations are required for the Lanczos algorithm to converge?

I am trying to find the n smallest eigenvalues and eigenvectors of a NxN SPD matrix using Lanczos method. What is the number of iterations usually required? I mean, does it scale as $O(N)$ or ...
7
votes
0answers
110 views

How do I find elements of an algebra which generate an algebra contained in a fixed subspace?

Suppose $V$ is a linear subspace of a finite dimensional $C^*$-algebra $A$. (Feel free to assume $A$ is a multi-matrix algebra over $\mathbb C$.) I would like to find $x \in V$ such that $\mathbb C ...
5
votes
2answers
880 views

Factorizing a block symmetric matrix

Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible. I would like to ...
4
votes
2answers
2k views

Is there some algorithms for solving non-linear matrix equations?

Is there some algorithms for solving non-linear matrix equations on field $\mathbb{C}$? Especially, solving polynomial nonlinear matrix equations. For instance, let $X$ be some matrix satisfying ...